Attracting much attention as a ceramic honeycomb filter for capturing particulate matter contained in an exhaust gas from a diesel engine and burning the particulate matter by a heater or a burner disposed at a desired position is a ceramic honeycomb filter constituted by a ceramic honeycomb structure having flow paths partitioned by porous cell walls with predetermined flow paths plugged, a gas mainly passing through the porous cell walls. As shown in FIG. 5(a) and FIG. 5(b), a ceramic honeycomb filter 50 is usually a cylinder having a circular or elliptical cross section, which has a structure comprising a porous ceramic honeycomb structure 51 having peripheral wall 51a and cell walls 51b for defining large numbers of flow paths 51c inside the peripheral wall 51a, and plugs 52a, 52b alternately sealing inlets 51d and outlets 51e of the flow paths 51c. 
As the characteristics of the ceramic honeycomb filter, a particulate-matter-capturing ratio, a pressure loss, and a particulate-matter-capturing time (time period until reaching a predetermined pressure loss from the start of capturing) are important. The particulate-matter-capturing ratio and the pressure loss are generally in a contradictory relation, and to achieve good balance between these contradictory characteristics, investigation has been being conducted so far on the control of the porosity, average pore diameter, pore diameter distribution on cell wall surfaces, etc. of the ceramic honeycomb filter. In addition, because the ceramic honeycomb filter is not only exposed to a high-temperature exhaust gas, but also heated by a heater or a burner for burning the captured particulate matter, it is required to have high thermal shock resistance to withstand severe conditions such as rapid temperature change. Thus, investigation has been being conducted to reduce its thermal expansion coefficient.
JP2002-326879A discloses a method for obtaining a high-porosity ceramic honeycomb structure by using a moldable material comprising foamed resin particles and resin powder, etc. This method provides a ceramic honeycomb structure having a porosity of 58-81% free from cracking by sintering, by adding the foamed resin particles without using a large amount of inflammable powder.
JP9-77573A discloses a honeycomb structure having a high capturing ratio, a low pressure loss and a low thermal expansion coefficient, which is formed by adding a foaming resin or an inflammable material to a cordierite-forming material, the honeycomb structure having a thermal expansion coefficient of 0.3×10−6/° C. or less, a porosity of 55-80%, and an average pore diameter of 25-40 μm, and pores exposed to cell wall surfaces being constituted by small pores of 5-40 μm and large pores of 40-100 μm, the number of the small pores being 5-40 times that of the large pores.
However, in the ceramic honeycomb structures described in JP2002-326879A and JP9-77573A, no consideration is made on the shapes of pores inside the cell walls, though their porosities are as high as 58-81% and 55-80%, respectively. Accordingly, it is difficult to achieve both high capturing ratio of particulate matter and low pressure loss.
When various pore-forming agents (foaming resin particles, inflammable materials, foamed resin particles, resin powder, etc.) described in JP2002-326879A and JP9-77573A are added to cordierite material powder to obtain a high-porosity cordierite ceramic honeycomb structure, the orientation of the cordierite material powder may be disturbed during extrusion. As a result, sufficient orientation of a cordierite crystal cannot be achieved, resulting in ceramic honeycomb structures having as large thermal expansion coefficients as more than 12×10−7/° C., and poor thermal shock resistance.
Development is recently conducted to put into practical use regenerable-with-catalyst ceramic honeycomb filters for continuously burning particulate matter in an exhaust gas by a catalytic reaction, which comprise ceramic honeycomb structures having cell walls with pores in which catalysts are supported. In such regenerable-with-catalyst ceramic honeycomb filters, various attempts have been conducted to increase the opening diameters and opening area ratios of pores in cell walls on the exhaust-gas-inlet side as described below, to increase the contact efficiency of particulate matter to the catalyst.
JP2002-309921A discloses an exhaust gas-cleaning apparatus comprising a porous ceramic whose pores have opening diameters of 30 μm or more in cell walls facing flow paths on the exhaust gas inlet side, and less than 30 μm in cell walls facing flow paths on the exhaust gas outlet side.
JP2002-349234A discloses a diesel exhaust gas-cleaning filter, wherein the total area of pores open to cell wall surfaces is 30% or more of the total surface area of cell walls, and wherein the total opening area of pores having diameters of 30 μm or more is 50% or more of the total opening area of all open pores.
JP2002-355511A discloses an exhaust gas-cleaning filter comprising a ceramic honeycomb structure and a catalyst supported on its cell walls, the cell walls having a porosity of 55-80%, and the percentage (%) of surface pores [expressed by B/A, wherein A is a total surface area, and B is a total area of pores, on a SEM photograph of a cell wall surface] being 20% or more.
JP2003-120256A discloses an exhaust gas-cleaning filter comprising a honeycomb structure having partition walls having large numbers of pores, and cells defined by partition walls, wherein the opening area ratio of pores having opening diameters of 10 μm or more is 20% or less of the total opening area ratio of pores open to the partition walls. This filter has a small number of small pores having opening diameters of 10 μm or less.
JP2003-40687A discloses a ceramic honeycomb structure having a porosity of 55-65% and an average pore diameter of 15-30 μm, wherein the total area of pores exposed to cell wall surfaces is 35% or more of the total surface area of cell walls.
However, even when such regenerable-with-catalyst ceramic honeycomb filter is used, the amount of particulate matter captured is more than that treated by a catalytic reaction because of low catalyst activity in an operation state at a low exhaust gas temperature, so that the particulate matter is accumulated in the pores of the cell walls, resulting in large pressure loss. Accordingly, JP2003-155919A proposes the forced regeneration of a ceramic honeycomb filter by adding a fuel to an exhaust gas upstream of the ceramic honeycomb filter when a large amount of particulate matter was accumulated.
However, when the regenerable-with-catalyst ceramic honeycomb filter is forcedly regenerated to prevent the increase of pressure loss due to the accumulation of particulate matter, the use of the ceramic honeycomb structures of JP2002-309921A to JP2003-40687A as described above, which have cell walls having pores with increased opening diameters and opening area ratios on the exhaust gas inlet side, not only provides a low capturing ratio at the start of capturing particulate matter, but also suffers pressure loss increasing as more particulate matter is captured.
Thus, in the conventional regenerable-with-catalyst ceramic honeycomb filters, whose pores open to porous cell walls have large opening diameters and opening area ratios, particulate matter is likely to easily pass through large pores without being captured, particularly at an operation stage at which the exhaust gas temperature is low, and at an initial stage with a small amount of particulate matter accumulated. Whenever the ceramic honeycomb filters are forcedly regenerated, the accumulated particulate matter is burned out, so that the filter returns to the original state. Accordingly, until a predetermined amount of particulate matter is accumulated in the pores of cell walls, a state of a low particulate-matter-capturing-ratio occurs repeatedly, with some particulate matter discharged. Once the particulate matter is accumulated in the pores of cell walls, the particulate-matter-capturing ratio is improved. However, filling the pores with particulate matter causes large pressure loss in the ceramic honeycomb filter.